2 We define a function (, ·) : R² × R² → R by taking 1 2. Let A denote the 2 × 2 matrix (х, у) — уТАх, for each pair of vectors x, y € R². (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a real number. However, we identify the 1 x 1 matrix with its single real entry.) (a) Does (·, ·) define an inner product on R²? If it does, prove it. If it does not, state which inner product conditions this function satisfies, and prove those. Also, provide counterexamples for each inner product condition that this function does not satisfy. (b) By replacing A with another matrix B different from A in only a single entry, define an inner product [, ] different from the function in part (a), and justify why this modified function is an inner product. (c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ·] that you chose in part (b). Justify that your chosen basis is orthonormal.
2 We define a function (, ·) : R² × R² → R by taking 1 2. Let A denote the 2 × 2 matrix (х, у) — уТАх, for each pair of vectors x, y € R². (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a real number. However, we identify the 1 x 1 matrix with its single real entry.) (a) Does (·, ·) define an inner product on R²? If it does, prove it. If it does not, state which inner product conditions this function satisfies, and prove those. Also, provide counterexamples for each inner product condition that this function does not satisfy. (b) By replacing A with another matrix B different from A in only a single entry, define an inner product [, ] different from the function in part (a), and justify why this modified function is an inner product. (c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ·] that you chose in part (b). Justify that your chosen basis is orthonormal.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Please show the steps clearly. (writting is not too messy). I'll leave a like for sure. Thank you.
![(1 2
2. Let A denote the 2 x 2 matrix
We define a function (, ) : R² × R² → R by taking
(x, y) = y" Ax,
for each pair of vectors x, y E R². (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a
real number. However, we identify the 1 x 1 matrix with its single real entry.)
(a) Does (, ·) define an inner product on R? If it does, prove it. If it does not, state which inner
product conditions this function satisfies, and prove those. Also, provide counterexamples for
each inner product condition that this function does not satisfy.
(b) By replacing A with another matrix B different from A in only a single entry, define an inner
product [, ] different from the function in part (a), and justify why this modified function is an
inner product.
(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ·] that you
chose in part (b). Justify that your chosen basis
orthonormal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c652b07-cde8-4c58-85c1-7bc870d165e6%2F87b3b5e8-fcd2-43cd-9aba-755b1da96c9e%2Fbr5sildm_processed.png&w=3840&q=75)
Transcribed Image Text:(1 2
2. Let A denote the 2 x 2 matrix
We define a function (, ) : R² × R² → R by taking
(x, y) = y" Ax,
for each pair of vectors x, y E R². (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a
real number. However, we identify the 1 x 1 matrix with its single real entry.)
(a) Does (, ·) define an inner product on R? If it does, prove it. If it does not, state which inner
product conditions this function satisfies, and prove those. Also, provide counterexamples for
each inner product condition that this function does not satisfy.
(b) By replacing A with another matrix B different from A in only a single entry, define an inner
product [, ] different from the function in part (a), and justify why this modified function is an
inner product.
(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ·] that you
chose in part (b). Justify that your chosen basis
orthonormal.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 6 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
